Given x = [0, 0.05, 0.1, 0.15, 0.20, ... , 0.95, 1] and f(x) = [1, 1.0053, 1.0212, 1.0475, 1.0841, 1.1308, 1.1873, 1.2532, 1.3282, 1.4117, 1.5033, 1.6023, 1.7083, 1.8205, 1.9382, 2.0607, 2.1873, 2.3172, 2.4495, 2.5835, 2.7183], write a Matlab script that computes the 1st and 2nd derivatives of O(h^2).

Compute the first partial derivatives of h(x,y,z)=(1+9x+4y)^z

Solve the differential equation 2x^2y"-x(x-1)y'-y = 0 using the Frobenius Method

Find all integer solutions to the equation:

a) 105x + 83y = 1

b) 105x + 83y = 8

3. In R⁴ , does the set {(1, 1, 1, 0,(1, 0, 0, 0),(0, 1, 0, 0),(0, 0, 1, 1)}, span R⁴? In other words, can you write down any vector (a, b, c, d) ∈ R⁴ as a linear combination of vectors in the given set ? Is the above set of vectors linearly independent ?

4. In the vector space P₂ of polynomials of degree ≤ 2, find explicitly a polynomial p(x) which is not in the span of the set {x + 2, x² − 1}.

5. Let S be the subspace of P₂ defined by S := {ax² + bx + 2a + 3b : a, b ∈ R}, for different choices of real numbers a and b (you don’t need to show here that S is indeed a subspace, and can assume. But is a good practice problem). Find a basis, and hence dimension for S.

Description: A terrible zombie apocalypse is ravaging across planet earth. The number of zombies is growing proportionally to the number of zombies present. After one month since the first infection, 100 million people had been infected and by the time another month had passed, there were 400 million infected.

1)  Construct a differential equation which models the number of zombies. Explain the reasoning behind how you constructed the model you chose to use. Discuss any potential shortcomings of your model. Describe any initial conditions associated with your model.

1. Give an example of a 3rd order nonlinear ordinary differential equation.

Let A be a m × n matrix with entries in R. Recall that the row rank of A means the dimension of the subspace in R^N spanned by the rows of A (viewed as vectors in Rⁿ), and the column rank means that of the subspace in R^m spanned by the columns of A (viewed as vectors in R^m).

(a) Prove that

n = (column rank of A) + dim S,

where the set S is the solution space of the homogeneous equation AX = 0, that is, S = {column vectors X : AX = 0} .

(b) Show that

row rank of A = column rank of A.

Suppose you have been asked to develop a simple model for the movement of ants in the presence of a food source. Discuss how you could simulate this model. Your answer should include any simplifying assumptions, some parameters that you’d need, and the interaction rules you would set up.

i. Define Fourier Series and explain it usefulness. At what instance can a function ?(?) be developed as a Fourier series.
ii. If ?(?)=12(?−?), find the Fourier series of period 2? in the interval (0,2?)

Show theoretically that least-squares fitting and Lagrange polynomial fitting yields the same result when there are 2 data points and x₁= 0.

Air enters the compressor of a gas-turbine plant at ambient conditions of 100 kPa and 25°C with a low velocity and exits at 1 MPa and 347°C with a velocity of 90 m/s. The compressor is cooled at a rate of 1500 kJ/min, and the power input to the compressor is 235 kW. Determine the mass flow rate of air through the compressor. The inlet and exit enthalpies of air are 298.2 kJ/kg and 628.07 kJ/kg.

The mass flow rate of air is  kg/s.

Let a,b be an element in the integers with a greater or equal to 1. Then there exist unique q, r in the integers such that b=aq+r where z less than or equal r less than or equal a+(z-1). Prove the Theorem.

Draw ? on a torus (or the schematic representation where opposite

sides of a rectangle are identified.)

1. What is a differential?

2. What is a differential equation?

3. Besides the fact that you might need the course to graduate, how might differential equations be useful to you in real life?